Evaluations of fuzzy quantities

This paper deals with the problem of evaluating a fuzzy quantity. Here we define “fuzzy quantity” any fuzzy set that may be non-normal or/and non-convex. What we mean as “evaluation problem”? We deal with a procedure that assigns to every fuzzy quantity a real number. In this case we deal of a particular fuzzy quantity defined as the union of two, or more, non-normal fuzzy numbers. This type of fuzzy set is the typical output of a fuzzy inference system. Looking to this particular aspect, we are interested in an evaluation proposal that is useful either in a defuzzification problem or in ranking procedures. We propose a definition which arises from crisp set theory: it is based on a particular fuzzy number evaluation, weighted average value (WAV) that works on alpha-cuts levels and depends on two parameters, a real number and an additive measure S; is connected with the optimistic or pessimistic point of view of the decision maker, S allows the decision maker to choose evaluations in particular subsets of the fuzzy number support, according to his preference. We show interesting properties of this proposal and how it is possible to use it even in more complex fuzzy sets that are always not normal and not convex. A fuzzy quantity has had a development even in other fields of research. One of this is the definition of “intuizionistic fuzzy set”, introduced by Atanassov for other types of problems, which have gained a wide recognition as a useful tool for modeling uncertain phenomena.